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replace checksoltuion by certify
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@ -5,6 +5,7 @@ using LinearAlgebra
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using SparseArrays
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using Dates
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using IntervalArithmetic
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using JuMP
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using Groups
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@ -14,7 +15,7 @@ using SymbolicWedderburn
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include("laplacians.jl")
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include("constraint_matrix.jl")
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include("sos_sdps.jl")
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include("checksolution.jl")
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include("certify.jl")
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include("1712.07167.jl")
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include("1812.03456.jl")
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168
src/certify.jl
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168
src/certify.jl
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@ -0,0 +1,168 @@
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function augment_columns!(Q::AbstractMatrix)
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for c in eachcol(Q)
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c .-= sum(c) ./ length(c)
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end
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return Q
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end
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function _fma_SOS_thr!(
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result::AbstractVector{T},
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mstructure::AbstractMatrix{<:Integer},
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Q::AbstractMatrix{T},
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acc_matrix=zeros(T, size(mstructure)...),
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) where {T}
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s1, s2 = size(mstructure)
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@inbounds for k = 1:s2
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let k = k, s1 = s1, s2 = s2, Q = Q, acc_matrix = acc_matrix
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Threads.@threads for j = 1:s2
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for i = 1:s1
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@inbounds acc_matrix[i, j] =
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muladd(Q[i, k], Q[j, k], acc_matrix[i, j])
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end
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end
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end
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end
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@inbounds for j = 1:s2
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for i = 1:s1
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result[mstructure[i, j]] += acc_matrix[i, j]
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end
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end
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return result
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end
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function _cnstr_sos!(res::AlgebraElement, Q::AbstractMatrix, cnstrs)
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StarAlgebras.zero!(res)
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Q² = Q' * Q
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for (g, A_g) in cnstrs
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res[g] = dot(A_g, Q²)
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end
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return res
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end
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function _augmented_sos!(res::AlgebraElement, Q::AbstractMatrix)
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A = parent(res)
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StarAlgebras.zero!(res)
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Q² = Q' * Q
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N = LinearAlgebra.checksquare(A.mstructure)
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augmented_basis = [A(1) - A(b) for b in @view basis(A)[1:N]]
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tmp = zero(res)
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for (j, y) in enumerate(augmented_basis)
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for (i, x) in enumerate(augmented_basis)
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# res += Q²[i, j] * x * y
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StarAlgebras.mul!(tmp, x, y)
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StarAlgebras.mul!(tmp, tmp, Q²[i, j])
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StarAlgebras.add!(res, res, tmp)
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end
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end
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return res
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end
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function compute_sos(A::StarAlgebra, Q::AbstractMatrix; augmented::Bool)
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if augmented
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z = zeros(eltype(Q), length(basis(A)))
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res = AlgebraElement(z, A)
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return _augmented_sos!(res, Q)
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cnstrs = constraints(basis(A), A.mstructure; augmented=true)
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return _cnstr_sos!(res, Q, cnstrs)
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else
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@assert size(A.mstructure) == size(Q)
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z = zeros(eltype(Q), length(basis(A)))
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_fma_SOS_thr!(z, A.mstructure, Q)
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return AlgebraElement(z, A)
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end
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end
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function sufficient_λ(residual::AlgebraElement, λ; halfradius)
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L1_norm = norm(residual, 1)
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suff_λ = λ - 2.0^(2ceil(log2(halfradius))) * L1_norm
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eq_sign = let T = eltype(residual)
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if T <: Interval
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"∈"
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elseif T <: Union{Rational,Integer}
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"="
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else # if T <: AbstractFloat
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"≈"
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end
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end
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info_strs = [
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"Numerical metrics of the obtained SOS:",
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"ɛ(elt - λu - ∑ξᵢ*ξᵢ) $eq_sign $(aug(residual))",
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"‖elt - λu - ∑ξᵢ*ξᵢ‖₁ $eq_sign $(L1_norm)",
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" λ $eq_sign $suff_λ",
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]
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@info join(info_strs, "\n")
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return suff_λ
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end
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function sufficient_λ(
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elt::AlgebraElement,
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order_unit::AlgebraElement,
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λ,
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sos::AlgebraElement;
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halfradius
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)
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@assert parent(elt) === parent(order_unit) == parent(sos)
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residual = (elt - λ * order_unit) - sos
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return sufficient_λ(residual, λ; halfradius=halfradius)
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end
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function certify_solution(
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elt::AlgebraElement,
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orderunit::AlgebraElement,
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λ,
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Q::AbstractMatrix{<:AbstractFloat};
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halfradius,
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augmented=iszero(aug(elt)) && iszero(aug(orderunit))
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)
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should_we_augment = !augmented && aug(elt) == aug(orderunit) == 0
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Q = should_we_augment ? augment_columns!(Q) : Q
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@time sos = compute_sos(parent(elt), Q, augmented=augmented)
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@info "Checking in $(eltype(sos)) arithmetic with" λ
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λ_flpoint = sufficient_λ(elt, orderunit, λ, sos, halfradius=halfradius)
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if λ_flpoint ≤ 0
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return false, λ_flpoint
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end
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λ_int = @interval(λ)
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Q_int = [@interval(q) for q in Q]
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check, sos_int = @time if should_we_augment
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@info("Projecting columns of Q to the augmentation ideal...")
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Q_int = augment_columns!(Q_int)
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@info "Checking that sum of every column contains 0.0..."
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check_augmented = all(0 ∈ sum(c) for c in eachcol(Q_int))
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check_augmented || @error(
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"Augmentation failed! The following numbers are not certified!"
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)
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sos_int = compute_sos(parent(elt), Q_int; augmented=augmented)
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check_augmented, sos_int
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else
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true, compute_sos(parent(elt), Q_int, augmented=augmented)
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end
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@info "Checking in $(eltype(sos_int)) arithmetic with" λ
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λ_certified =
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sufficient_λ(elt, orderunit, λ_int, sos_int, halfradius=halfradius)
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return check && inf(λ_certified) > 0.0, inf(λ_certified)
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end
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@ -1,77 +0,0 @@
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using IntervalArithmetic
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IntervalArithmetic.setrounding(Interval, :tight)
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IntervalArithmetic.setformat(sigfigs=12)
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function fma_SOS_thr!(result::AbstractVector{T}, pm::AbstractMatrix{<:Integer},
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Q::AbstractMatrix{T}, acc_matrix=zeros(T, size(pm)...)) where T
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s1, s2 = size(pm)
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@inbounds for k in 1:s2
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let k=k, s1=s1, s2=s2, Q=Q, acc_matrix=acc_matrix
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Threads.@threads for j in 1:s2
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for i in 1:s1
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@inbounds acc_matrix[i,j] = muladd(Q[i, k], Q[j, k], acc_matrix[i,j])
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end
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end
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end
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end
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@inbounds for j in 1:s2
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for i in 1:s1
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result[pm[i,j]] += acc_matrix[i,j]
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end
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end
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return result
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end
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function compute_SOS(pm::AbstractMatrix{<:Integer}, Q::AbstractMatrix)
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result = zeros(eltype(Q), maximum(pm));
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return fma_SOS_thr!(result, pm, Q)
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end
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function compute_SOS(RG::GroupRing, Q::AbstractMatrix{<:Real})
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result = compute_SOS(RG.pm, Q)
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return GroupRingElem(result, RG)
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end
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function compute_SOS_square(pm::AbstractMatrix{<:Integer}, Q::AbstractMatrix)
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result = zeros(eltype(Q), maximum(pm));
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for i in 1:size(Q,2)
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GroupRings.fmac!(result, view(Q,:,i), view(Q,:,i), pm)
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end
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return result
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end
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function compute_SOS_square(RG::GroupRing, Q::AbstractMatrix)
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return GroupRingElem(compute_SOS_square(RG.pm, Q), RG)
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end
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function augIdproj(Q::AbstractMatrix{T}) where T
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result = zeros(T, size(Q))
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l = size(Q, 2)
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Threads.@threads for j in 1:l
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col = sum(view(Q, :,j))/l
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for i in 1:size(Q, 1)
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result[i,j] = Q[i,j] - col
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end
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end
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return result
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end
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function augIdproj(::Type{Interval}, Q::AbstractMatrix{T}) where {T<:Real}
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result = zeros(Interval{T}, size(Q))
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l = size(Q, 2)
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Threads.@threads for j in 1:l
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col = sum(view(Q, :,j))/l
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for i in 1:size(Q, 1)
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result[i,j] = @interval(Q[i,j] - col)
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end
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end
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check = all([zero(T) in sum(view(result, :, i)) for i in 1:size(result, 2)])
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return result, check
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end
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